Question 8
Assume that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the following condition:
For every $x, y \in \mathbb{R}$ such that $(f(x)+y)(f(y)+x)\gt 0$, we have $f(x)+y=f(y)+x$.
Prove that $f(x)+y \leqslant f(y)+x$ whenever $x\gt y$.
For every $x, y \in \mathbb{R}$ such that $(f(x)+y)(f(y)+x)\gt 0$, we have $f(x)+y=f(y)+x$.
Prove that $f(x)+y \leqslant f(y)+x$ whenever $x\gt y$.